WordGPT

Solve x² + 11x - 10 = 0

The solutions of x² + 11x - 10 = 0 are x = (-11 + √161)/2 or x = (-11 - √161)/2, that is x ≈ 0.84428877 and x ≈ -11.844289.

x² +x += 0
x2+11x−10=0⇒x=−11+1612, −11−1612x^2 + 11x - 10 = 0 \Rightarrow x = \frac{-11 + \sqrt{161}}{2},\ \frac{-11 - \sqrt{161}}{2}
Discriminant D = 161 · x ≈ 0.844289, -11.8443

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

Explain it step by step in chatMake a short video of it
Graph of y = x² + 11x - 10-14-12-10-8-6-4-22-40-202040x = 0.84428877x = -11.844289vertexy = x² + 11x - 10
Graph of y = x² + 11x - 10. The curve crosses the x-axis at x = 0.84428877 and x = -11.844289; the vertex is at (-5.5, -40.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = 11, c = -10.

    x2+11x−10=0x^2 + 11x - 10 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=112−4⋅1⋅(−10)=121+40=161D = 11^2 - 4\cdot1\cdot(-10) = 121 + 40 = 161
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−11±1612x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{161}}{2}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=−11+1612orx=−11−1612x = \frac{-11 + \sqrt{161}}{2}\quad\text{or}\quad x = \frac{-11 - \sqrt{161}}{2}
  6. 6

    As decimals: x ≈ 0.84428877 and x ≈ -11.844289.

  7. 7

    Check numerically: for x ≈ 0.84428877 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 11x - 10 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{161}}{2},\ \frac{-11 - \sqrt{161}}{2} \]
\[ x^2 + 11x - 10 = 0 \]
\[ D = 11^2 - 4\cdot1\cdot(-10) = 121 + 40 = 161 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{161}}{2} \]
\[ x = \frac{-11 + \sqrt{161}}{2}\quad\text{or}\quad x = \frac{-11 - \sqrt{161}}{2} \]

The parabola y = x² + 11x - 10

xy
-8.5-31.25
-7.5-36.25
-6.5-39.25
-5.5-40.25
-4.5-39.25
-3.5-36.25
-2.5-31.25

Questions

What are the solutions of x² + 11x - 10 = 0?

The solutions of x² + 11x - 10 = 0 are x = (-11 + √161)/2 or x = (-11 - √161)/2, that is x ≈ 0.84428877 and x ≈ -11.844289.

Does x² + 11x - 10 = 0 factor?

Not over the integers, because the discriminant 161 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 11x - 10?

(-11/2, -161/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -11, product = c/a = -10.

Similar equations

    Solve x² + 11x - 10 = 0: x = (-11 + √161)/2 or x = (-11 - √161)/2 (steps) | WordGPT